Ship motion control based on AMBPS-PID algorithm
Wang, Le; Wu, Qing; Liu, Jialun ; Li, Shijie; Negenborn, Rudy DOI
10.1109/ACCESS.2019.2960098 Publication date
2019
Document Version Final published version Published in
IEEE Access
Citation (APA)
Wang, L., Wu, Q., Liu, J., Li, S., & Negenborn, R. (2019). Ship motion control based on AMBPS-PID algorithm. IEEE Access, 7, 183656-183671. https://doi.org/10.1109/ACCESS.2019.2960098
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Ship Motion Control Based
on AMBPS-PID Algorithm
LE WANG 1, QING WU 1, JIALUN LIU 2,4, SHIJIE LI 1, AND RUDY R. NEGENBORN 3,4
1School of Logistics Engineering, Wuhan University of Technology, Wuhan 430063, China 2National Engineering Research Center for Water Transport Safety, Wuhan 430063, China
3Department of Maritime and Transport Technology, Delft University of Technology, 2628CD Delft, The Netherlands 4Intelligent Transportation Systems Research Center, Wuhan University of Technology, Wuhan 430063, China
Corresponding author: Jialun Liu (jialunliu@whut.edu.cn)
This work was supported in part by the National Key Research and Development program of China under Grant 2018YFB1601505, in part by the National Natural Science Foundation of China under Grant 51709217, in part by the Research on Intelligent Ship Testing and Verification under Grant 2018473, in part by the Natural Science Foundation of Hubei Province under Grant 2018CFB640, in part by the State Key Laboratory of Ocean Engineering (Shanghai Jiao Tong University) under Grant 1707, in part by the Fundamental Research Funds for the Central Universities under Grant WUT2018IVA034 and Grant WUT2018IVB079, in part by the China Scholarship Council under Grant 201806950096, and in part by the Double First-rate Project of Wuhan University of Technology.
ABSTRACT Intelligent motion control is one of the key technologies of ships. This paper studies the application of Adaptive Mutation Beetle Particle Swarm (AMBPS)-PID algorithm in ship motion control. Firstly, the ship MMG model is established. Then the BAS algorithm is introduced, and AMBPS algorithm is improved and designed on this basis. Secondly, ship heading and path following controllers are designed according to the algorithm, and rudder turning rate constraint is introduced to limit the rudder angle. Thirdly, through the test function effect analysis of AMBPS and other similar algorithms, the improved effect of this algorithm is verified. Finally, from manual tuning PID parameters to off-line and on-line optimizing parameters based on AMBPS algorithm, the optimal control parameters are obtained step by step, and the optimal heading and path following simulation results are achieved. Compared with the results of traditional PID, AMBPS-PID algorithm has a better adaptive control effect on ship motion control, reduces the error of manual tuning parameters and improves efficiency.
INDEX TERMS Motion control, heading control, path following, AMBPS-PID algorithm.
I. INTRODUCTION
The main problems of ship motion control are the uncer-tainty of ship dynamics, random environmental disturbance and inaccuracy of measurement information [1]. For these reasons, researchers have carried out various control meth-ods to effectively reduce the impact of these reasons, such as PID (proportion, integral and derivative), fuzzy control, predictive control, sliding mode control and other basic rithms or their improved algorithms. Among these algo-rithms, the PID controller is the most widely used controllers with simple and fast characteristics. It is often combined with algorithms such as predictive algorithm and fuzzy control to control ship motion [2]–[4]. But there are still some short-comings for the PID algorithm of a complex control system. One of them is that the fixed parameters of the PID controller The associate editor coordinating the review of this manuscript and approving it for publication was Yilun Shang .
make the algorithm unable to satisfy the precise control of a time-varying system, so the relevant scholars applied the intelligent optimization algorithm to the optimization of the parameters of the PID control. For example, ant colony (AC) algorithm [5], genetic algorithm (GA) [6], [7], particle swarm optimization (PSO) algorithm [8], neural network (NN) algo-rithm [9]–[12] and bacterial foraging optimization (BFO) algorithm [13] are applied to modify the PID parameters to adapt to the different operating conditions of ships.
To sum up, it is a common method to optimize the parame-ters of the PID algorithm based on the intelligent optimization algorithm. However, the complexity of the intelligent opti-mization algorithm leads to a large amount of calculation and low efficiency. Beetle Antennae Search (BAS) algorithm is a kind of bionic optimization algorithm, which is proposed in 2017. It has the advantages of low computational com-plexity and fast speed [14]. In recent two years, the research of the BAS algorithm has developed to BAS swarm [15] or This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see http://creativecommons.org/licenses/by/4.0/
combined with PSO to improve the efficiency and effec-tiveness of optimization [16], [17]. Based on this, the paper introduces the mutation factor into the control algorithm combining BAS and PSO, and then acts on the PID control. The aim is to quickly find the optimal control parameters in the control algorithm by using a new optimization algorithm instead of manual experience.
The main contributions of this article are as follows: 1) The rudder turning rate control is introduced into the
motion controller to make the control results more in line with the actual ship situation.
2) Adaptive Mutation Beetle Particle Swarm (AMBPS) algorithm is designed to replace manual experience with a new optimization algorithm. It can quickly find the optimal control parameters in the control algorithm and reduce the dynamic changes of ship control. Firstly, the special parameters can be excluded by manual tun-ing. Then, the algorithm variables are analyzed for a single heading angle, and the optimal PID parameters under different headings are obtained.
3) In order to realize the self-adaptability and high effi-ciency of the algorithm, the optimal frequency of the PID parameters is simulated and analyzed. Finally, the heading angle control and path following under the optimal parameters are realized.
Following this introduction, this paper is mainly divided into six sections. Maneuvering Modeling Group (MMG) model of the ship is built in SectionII. SectionIIIdescribes the AMBPS algorithm. Then, the ship motion controller based on AMBPS-PID is proposed in SectionIV. SectionV
elaborates on the simulation results and the analysis of the algorithm. Finally, this paper makes the summaries of the results and put forward to the future research targets in Section VI. Appendix A is the symbols meaning table, and AppendixBis the comparison of testing results of basic AMBPS algorithm and other algorithms.
II. MATHEMATICAL MODELING MODEL OF A SHIP
The accuracy of the ship model affects the accuracy of motion control. Generally, ship motion models can be divided into two categories, one is the hydrodynamic model, and the other is the responsive model. The hydrodynamic model is usually divided into the Abkowitz model and the MMG model [18]. This paper chooses the MMG model because it emphasizes the physical meaning of each hydrodynamic derivative, and takes into account the interaction between ship, propeller and rudder.
The space-fixed coordinate system o0 − x0y0z0 and the moving ship fixed coordinate system o−xyz, shown in Fig.1. The fixed coordinate system is set at the center of gravity of the ship.
Among them, ψ is defined as the angle between x0and
xaxes. In this paper we call it heading angle. u and vmdenote
the velocity components in x and y directions respectively, and U is the total velocity. δ represents rudder angle, β is
FIGURE 1. Coordinate systems.
defined as the drift angle at midship position. The maneuver-ing motion of a ship can be described by the MMG model as Eq.1[21]. XH+ nR X i=1 XRi + nP X j=1 XPj =(m + mx) · u −(m + my) · vm− xG· m · r2 YH+ nR X i=1 YRi =(m + my) · ˙vm +(m + mx) · u · r + xG· m · ˙r NH + nR X i=1 NRi =(IzG+ xG2· m + Jz) · ˙r + xG· m(˙vm+ u · r), (1) where H , R and P means hull, rudder and propeller, respec-tively. The i and j refers to each rudder and each propeller respectively, nR are np are the numbers of rudders and
pro-pellers, xG is the longitudinal coordinate of center of
grav-ity of ship, X , Y and N represents the longitudinal force, the transverse force and the transverse moment of the whole, respectively.The expression for the hull XH, YH and NH are
as Eq.2. XH =(1/2) · ρ · LPP· d · U2· XH0 (v 0 m, r 0 ) YH =(1/2) · ρ · LPP· d · U2· YH0(v 0 m, r 0) NH =(1/2) · ρ · LPP· d · U2· NH0 (v 0 m, r 0 ), (2)
where the XH0 is expressed as the longitudinal force coeffi-cient, YH0 is the lateral force coefficient of the ship, NH0 is expressed as the yaw moment coefficient. The effective rud-der forces and moments acting on the rudrud-der are as Eq.3.
XR= −(1 − tR) · FN·sinδ YR= −(1 − aH) · FN·cosδ NR= −(xR+ aH· xH) · FN·cosδ, (3)
FIGURE 2. BAS and BASS swarm principle.
where the FN is the rudder normal force, aH is the rudder
force multiplier, xRis the longitudinal coordinate of rudder
position(=−0.5LPP). xH is the longitudinal coordinate of the
acting point of the additional lateral force. The surge forces due to the propeller are as Eq.4.
XP= −(1 − tP) · T (4)
In addition to the meanings of the symbols indicated above, the other symbols of Eq. 1, 2, 3, 4 meaning are shown in AppendixA.
III. AMBPS ALGORITHM RESEARCH
The BAS [14] is an algorithm developed for the beetle forag-ing principle. The basic principle of BAS is shown in Fig. 2a. Through basic research of the BAS algorithm these years, it is found that the basic BAS algorithm is easy to fall into local optimum, and the search results depend greatly on the setting of the initial position [15], [16]. Therefore, the research can be extended from an individual to a swarm, namely a BAS swarm algorithm. The principle of BAS swarm is shown in Fig. 2b. From the schematic diagram of the BAS swarm algorithm, it can be found that each beetle is equivalent to one particle, so it can be further optimized by imitating the principle of PSO.
PSO is an evolutionary algorithm that uses massless par-ticles to simulate birds in a flock [19]. It seeks the optimal solution through cooperation and information sharing among individuals in a swarm. Similar to the PSO algorithm, a beetle can act as a particle. Each particle has only two attributes:
speed and position. Among them, speed represents the level of movement, and position represents the direction of move-ment. The distance and direction of beetle movement are determined by the speed and fitness of antenna detection.
Based on the above, it is assumed that there are n beetles
X =(X1, X2, . . . , Xn) in N -dimensional space. The position
of the ith beetle is expressed as Xi = (xi1, xi2, . . . , xiN)
and the speed is expressed as Vi = (vi1, vi2, . . . , viN).
The individual extreme of beetles is expressed as Pbesti =
(Pbesti1, Pbesti2, . . . , PbestiN), and the global extreme is
expressed as Gbesti = (Gbesti1, Gbesti2, . . . , GbestiN).
So the speed Vi and original position ˜Xi of the beetle as a
particle are as Eq.5aand Eq.5b.
Vi(k) =ω · Vi(k) + c1· r1·(Pbesti(k − 1) − ˜Xi(k − 1))
+ c2· r2·(Gbesti(k − 1) − ˜Xi(k − 1)), (5a)
˜
Xi(k) = ˜Xi(k − 1) + Vi(k), (5b)
Among them,ω uses ‘‘Linearly Decreasing Weight (LDW)’’ to adjust itself according to the number of iterations, that is ω(k) = ωmin+(ωmax−ωmin) · (K − k)/K. ωmax andωmin
are the maximum and minimum weight coefficients, K is the maximum number of iterations, k is the current number of iterations.
According to the characteristics of beetle’s two anten-nae, its movement direction can be further judged. Two
N-dimensional vectors XL and XR are defined as the left
and right antenna coordinates of beetles respectively, shown as Eq.6. Antennae orientation can be expressed as random vector Dir(k) = rand (N, 1), and Dir is normalized to ˜D(k) =
Dir(k)/norm(Dir(k)). Besides, D0= Lstep/b2.
XLi(k) = ˜Xi(k) + D0· ˜D(k)/2
XRi(k) = ˜Xi(k) − D0· ˜D(k)/2 )
(6)
Xirepresents the beetle centroid coordinates and is also the
updated position coordinates of the beetle. The calculation method of Xi is shown in Eq.7, then the initial cost value is
calculated according to Xi, which is regarded as the current
optimal fitness function value fbest.
Xi(k) = ˜Xi(k) − Lstep· normal(XLi(k) − XRi(k)). . .
·sign(f (XLi(k)) − f (XRi(k)) (7)
The optimal fitness function is calculated according to the updated beetle position Xi(k) in each iteration cycle. Then
compare and update the current best parameters Xbest(k) and fbestaccording to Eq.8.
fbest(k) = f |Xi(k)
Xbest(k) = Xi(k)
)
, f |Xi(k)< fbest (8)
Update Lstep and D0. Lstep and D0 can be updated by the proportional method or other methods. If the optimal fitness value decreases gradually, only the beetle position
Xi(k) needs to be updated. On the contrary, the Lstepand D0 should be attenuated until the control is over or the optimal
fitness value set is satisfied. Because it means that the current
Lstepand D0can not get the optimal position, shown in Eq.9.
Lstep(k) = b1· Lbest(k − 1) D0(k) = Lstep(k − 1)/b2
)
, f |Xi(k)< fbest (9)
As the search proceeded, the difference between individual positions of beetles gradually decreased and the concentra-tion degree increased. To further reduce the probability of the algorithm falling into local optimum, an adaptive mutation operator is introduced in this paper. The basic principles are as follows: Firstly, the variance of population fitness function and spatial location aggregation degree are analyzed, then the diversity of the population is increased by judging whether an adaptive mutation is needed in threshold selection, and finally, the global optimization is realized.
At the current kth iteration, the fitness function is fi(k) = f |Xi(k), and the average fitness function of the population is
fave(k), then the fitness variance of the population is defined
as Eq.10. In this paper,2(k) is used to reflect the aggregation degree of a single beetle in the beetle population.
2(k) = n X i=1 f i(k) − favg(k) ˜ f(k) 2 , (10)
where, ˜f(k) = max{max{|fi(k) − favg(k)|}, 1}, i ∈ [1, n], its
function is to limit the size of2(k).
In addition, the degree of aggregation between beetles can also be expressed as Eq. 11by τ(k). The larger the τ(k) , the lower the probability of mutation.
τ(k) = max 1≤i≤nkXi (k) − Pbesti(k)k max k n max 1≤i≤nkXi(k) − Pbesti(k)k o (11)
When2(k) tends to zero and τ(k) decreases, the algorithm may fall into premature convergence. Define mutation prob-ability pm(k) as Eq.12. pm(k) = ( e−τ(k)/5, 2(k) < 20 0, 2(k) ≥ 20 (12) 20 is a threshold set according to the actual situation, usually close to 0. According to this probability, the mutation sequence is generated and the extreme of each individual is adjusted, such as Eq. 13. Among them, prand is a random
number between 0 and 1, andη is a N-dimensional random vector obeying (0, 1) normal distribution.
Pbesti(k) =
(
Pbesti(k) · (1 + 0.5η), prand(k)< pm(k) Pbesti(k), prand(k) ≥ pm(k)
(13)
IV. DESIGN OF SHIP MOTION CONTROLLER BASED ON AMBPS-PID
A. SHIP HEADING CONTROLLER
The aim of heading control is to calculate the rudder turning rate command of each step, and then input it into the ship
FIGURE 3. Heading tracking schematic.
motion model to get the next step heading angles, so as to gradually realize the target heading. Based on this, the control variables of heading control in this paper is ˙δEand the control
output isψ. In the o0− x0y0z0coordinate system used in this paper, Direction of vectors−−−−→DoDo+1(o = 1, 2, . . .) represents the target headingψreq. Ps(ys, xs) is the current position of the
ship, as shown in Fig.3.
Heading tracking aims to make the current heading angle of the ship follow the target heading angle. The heading errorψerror at tstep(indicated byλ) is as Eq.14.
ψerror(λ) = ψ(λ) − ψreq(λ). (14)
Based on the ψerror and the basic principle of the
PID algorithm, the next time step of the rudder command is shown as Eq.15. δE(λ + 1) = KPψerror(λ) + KI K X j=1 ψerror(λ) + KD(ψerror(λ) − ψerror(λ − 1)) (15)
According to the type of ship, rudder angle has a range of constraints, set to [δELmax, δERmax]. It needs to be judged after
getting theδE(λ + 1), shown as Eq.16.
δE(λ + 1) =(δELmax, δE
(λ + 1) > δELmax
δERmax, δE(λ + 1) > δERmax
(16) In practical applications, especially for large-scale ships, because of its huge mechanical structure, the rudder rotation has a certain speed. The rudder angle commands given by the control decision system need to be accumulated over a certain period time before they can be realized. Most of the current studies ignore the rudder turning time. It is directly considered that the rudder angle has reached the required target rudder angle in the next step, so it has a certain error compared with the actual situation. This article introduced the rudder turning rate constraint for the object(7 meters ship). It controls the rudder turning rate in each execution. Rudder turning rate control principle is as Eq.17.
˙
FIGURE 4. The principle of LOS navigation.
among them, tstep is the time step, ˙δ(λ + 1) is the next
time step rudder turning rate. Besides, rudder turning rate is constrained by left and right rudder turning rate, set to [ ˙δELmax, ˙δERmax]. The ˙δ(λ + 1) is obtained according to the
constraints, shown as Eq.18. ˙ δE(λ + 1) =( ˙δELmax, ˙δE (λ + 1) > ˙δELmax ˙ δERmax, ˙δE(λ + 1) > ˙δERmax (18) Then the rudder angle obtained through integral after judg-ing the rudder turnjudg-ing rate. It can be used on the MMG model to make a series calculation.
B. SHIP PATH FOLLOWING CONTROLLER
LOS navigation is a method to simulate the sight of experi-enced sailors to achieve convergence to the desired path [20]. In this paper, LOS guidance law is used to track the LOS angle, and then the track tracking results are obtained. The goal of path following is to calculate the rudder turning rate command controlled by each step and then input it into the ship motion model. Based on the principle of LOS navigation, it can gradually track the heading and then track the target path. In this paper, the control variable is ˙δEand the control
output isψ and Ps. The principle of LOS navigation is shown
in Fig.4.
As shown in the Fig.4, path points Pp(p = 0, 1, 2, . . .)
constitute the desired path, and the line between two adjacent points PpPp+1is a straight line. Among them, the straight line formed by Pp(yp, xp)Pp+1(yp+1, xp+1) is the current desired path. Plos(ylos, xlos) is the closest point between the Los circle
and the next point in the intersection of the target path.ψlosis
defined as an angle within 0−360 degrees with the northward direction as the reference point. It can be seen that no matter where the ship is currently located in Ps(ys, xs), the solution
formula of Plos(λ)(ylos(λ), xlos(λ)) is as Eq. 19a. Then the
calculation method ofψlos(λ) is shown in Eq. 19b. Plos = xlos(λ) − xp ylos(λ) − yp = xp+1− xp yp+1− yp (ylos(λ) − ys(λ))2+(xlos(λ) − xs(λ))2= R2AC, (19a) ψlos(λ) = arctan ylos(λ) − ys(λ) xlos(λ) − xs(λ) , (19b)
Besides, it can be seen from Fig.4that there is one or two intersection points, and there may be no real solution when the ship deviates from the desired path. To avoid this, Rlos
is usually set toζ times of captain Lpp. In order to ensure
the adaptability of ship path following, LOS circle is set as a dynamic circle. Rloscan be defined as:
Rlos(λ) = dl(λ) + ζLpp (20)
among them, the vertical distance dl(λ) from the ship’s posi-tion point to the target path is:
da= q (ys(λ) − yp+1)2+(xs(λ) − xp+1)2 db= q (yp+2− ys(λ))2+(xp+2− xs(λ))2 dc= q (yp+2− yp+1)2+(xp+2− xp+1)2 (21a) dl(λ) = q d2 a−((dc2− db2+ da2)/(2dc))2 (21b)
When the ship approaches the target point Pp+1(yp+1, xp+1), the target path needs to be switched from PpPp+1 to
Pp+1Pp+2. When the ship enters the acceptance circle with
Pp+1 as the center and RAC as the radius, that is, (ys(λ) − yp+1)2+(xs(λ) − xp+1)2≤ R2AC, the target path is switched. In order to ensure that point Plos(λ) may be located at
inflec-tion point Pp+1, this paper defines RAC as: RAC = Rlos(λ).
Finally, letψreq =ψlos. Based on SectionIV-A, the
rud-der turning rate command can be obtained to achieve path following.
C. SHIP MOTION CONTROLLER BASED ON AMBPS-PID ALGORITHM
This paper combines the AMBPS algorithm with the PID algorithm to design the ship motion controller. The flow chart of the algorithm is shown in Fig. 5. As shown in Fig. 5, the k step variable values are calculated by the k − 1 step variable values. In this paper, the control variables are three parameters of PID: KP, KI, KD. Then the current global
opti-mal fitness function value is judged and stored, and self-adaptively judge whether the control variables need to be mutated according to the control situation. For ship heading control and path following control, the fitness function f (x) is shown in Eq.22and 23, respectively. As shown in Eq.22
and 23, the fitness function is set as the tracking error of heading or path. The error judgment standard uses Root Mean Square Error(RMSE) and Mean Absolute Deviation(MAD). RMSE is used to measure the deviation between the observed value and the true value. It is very sensitive to the extremely large and very small errors in the results. Since MAD is abso-lutely quantized, the positive and negative phase cancellation will not occur. Therefore, the average absolute error can better reflect the actual situation of the error.
f1(x) = PK k=1|ψerror|
K , Minimum MAD as standard.
s
PK
k=1ψerror2
K , Minimum RMSE as standard.
FIGURE 5. Flow charts of AMBPS-PID algorithm. f2(x) = PK k=1|dl|
K , Minimum MAD as standard.
s
PK
k=1dl2
K , Minimum RMSE as standard.
(23) Besides, to analyze the application effect of the algo-rithm, the off-line and on-line controllers of ship motion are designed in this paper, as shown in Fig. 6. The off-line controller is to automatically find the most precise
FIGURE 6. Flow charts of off-line and on-line controllers.
PID parameters through multiple cycles, which is suitable for single heading control. The on-line controller is a self-adaptive controller that automatically updates and optimizes the PID parameters in a single or k steps. The aim is to obtain the optimal PID parameters in the current or future k steps.
V. SIMULATION AND COMPARISON
A. SIMULATION AND COMPARISON OF AMBPS ALGORITHM
To ensure the effectiveness of the AMBPS optimization algo-rithm, this paper takes six classical global optimal test func-tions as examples to compare and analyze the effectiveness of the AMBPS algorithm. Six classical global optimal functions have their characteristics, which can effectively test whether the algorithm can jump out of local optimum.
• The Rastrigrin function is characterized by a large num-ber of deep local optima arranged by sinusoidal inflec-tion points.
• The product term between variables of the Griewank function has strong interaction and is a multi-modal function.
• The Michalewicz function is a standard benchmark function with several local minimum values and plane regions.
• The Goldstein-Price function has several local minimum values.
• Ackley function is characterized by an almost flat region modulated by cosine waves forming holes or peaks, which make the surface undulate and uneven.
• The characteristic of the Schaffer function is that the local optimal value is located on the concentric circle near the global optimal value, while the global optimal value is located in a very narrow concave region. This paper designs AMBPS to compare with BAS [14], BAS Swarm, PSO ,and BAS-PSO [16] algorithms. When the
FIGURE 7. Tracking RMSE and MAD for Ship single target heading angle.
TABLE 1. Main particulars of the 7m KVLCC2 ship.
Titer =1000 and n = 10, the average error of 10 times calcu-lations and the comparison results are detailed in AppendixB. It can be seen from AppendixBthat the AMBPS algorithm is more stable than other algorithms in the case of ensuring high precision, and can effectively avoid falling into local opti-mum. To optimize the parameters of the PID algorithm in this paper, the adaptive and stable characteristics are the preferred standard for selecting optimization algorithms. Therefore, the shortcomings of AMBAS sometimes having a slight lack of optimization speed can be ignored.
B. SIMULATION RESULTS OF THE SHIP HEADING CONTROL WITH MANUAL SEARCHING
PID PARAMETERS
In this paper, the 7m KVLCC2 ship is selected as the simu-lation object. The basic parameters of the ship are shown in Table1. Detailed parameters can be found in reference [21].
According to the actual ship’s navigation situation, the range of theψ in this paper is specified as [−180◦, 180◦], and any target heading angle ψreq is selected to
simu-late in this range. For a single heading angle, we choose −30◦, −60◦, −100◦, 30◦, 60◦, 100◦ as examples to test. Firstly, PID parameters are searched manually, and the
TABLE 2.RMSE and MAD values for tracking different heading angles (Manual).
control errors under different parameters are obtained, as shown in Fig.7.
In Fig.7, the red lines represent the changes of the values of the PID parameters [KP, KI, KD], while the other lines
represent the MAD and RMSE values with the changes of the PID parameters under the different heading angle targets. It can be concluded from Fig.7: The change of KIhas a great
influence on the error, both MAD and RMSE values. Only when KIapproaches 0, the influence on the error is extremely
small, so KI is directly set to 0 in this paper. This is the only
certain PID parameter value that can be obtained by manual adjustment. Now the optimal PID parameter can be roughly obtained as [KP, KI, KD] = [5, 0, 20].
1) TRACKING A SINGLE HEADING ANGLE
Based on the manually adjusted PID parameter values [KP, KI, KD] = [5, 0, 20], the tracking step is set nStep =
200. Taking −30◦, −60◦, −100◦, 30◦, 60◦, 100◦as an exam-ple, the tracking result is shown in Fig. 8. Fig. 8a shows the ship trajectories under different headings, and 8b shows the changes of PID parameters, the tracking heading angle, the tracking error per tstepand the rudder angle respectively.
The heading tracking error MAD and RMSE results are shown in Table2.
2) TRACKING TIME-VARYING HEADING ANGLES
Based on the manually adjusted PID parameter values [KP, KI, KD] = [5, 0, 20], the tracking step is set
FIGURE 8. Tracking different heading angles (Manual).
TABLE 3. MAD and RMSE values for tracking time-varying heading angles (ψreq=Asin(0.01t), Manual).
nStep = 700. Similarly, the heading angle with sinusoidal function in [−180◦, 180◦] is chosen as the target heading angle, as shown in Fig.9. Fig. 9a shows the ship trajectories under different headings, and 9b shows the changes of PID parameters, the tracking heading angle, the tracking error and the rudder angle respectively. The MAD and RMSE for tracking time-varying heading angles are obtained, as shown in Table3.
From the above results, it can be concluded that the manual adjustment of parameters can achieve better tracking effect, but it needs to keep trying to find the better parameters. This method is inefficient and uncertain whether it is the optimal value, which can be used for initial judgment. Especially for the parameter KI in this paper, the optimal result can be
obtained by manual adjustment.
FIGURE 9. Tracking different time-varying heading angles (Manual).
C. SIMULATION RESULTS OF THE SHIP HEADING CONTROL AND PATH FOLLOWING WITH
AMBPS-PID ALGORITHM
1) TRACKING A SINGLE HEADING ANGLE (OFF-LINE CONTROLLER)
The basic control parameter values of the AMBPS-PID algo-rithm are shown in Table4. According to SectionIII, it can be found that the basic variables that may affect the results are n, Lstep, D0 and Vm. Using the control variable method,
setting the search times Titer = 100. Based on the Table4, the influence of variables is analyzed as follows.
1) With MAD optimum as the standard, the optional angle can be selected for simulation. There we choose 60◦as an example, the tracking error and the optimal PID parameters under different variables values are obtained as shown in Table5.
Optimum parameters are obtained from Table 5 are
n = 10, D0 = 0.99, Lstep = 1, Vm = 5. Under
TABLE 4. AMBPS-PID algorithm basic parameter values.
TABLE 5. Tracking error and the optimal PID parameters under different variable values (ψreq=60◦, minimum MAD).
TABLE 6. Tracking different heading angle errors and PID parameters (AMBPS-PID, minimum MAD).
tracked as shown in Fig.10and Table6. Fig. 10a shows the ship trajectories of different headings, and Fig. 10b shows the PID parameters, the tracking heading angle, search for the minimum error and the rudder angle change respectively.
2) With RMSE optimum as the standard, the tracking error and the optimal PID parameters under different variable values are obtained as shown in Table7.
Similarly, the optimum parameters are n = 10, D0 = 0.99, Lstep=1, Vm = 5. the other angle can be tracked
as shown in Fig.11and Table8.
From Table 6 and Table 8, it can be concluded that the optimal PID parameters for tracking any heading angle are slightly different, basically fluctuate around [KP, KI, KD] =
[10, 0, 40], both of MAD or RMSE. As can be seen from
FIGURE 10. Tracking different heading angles (AMBPS-PID, minimum MAD).
Fig. 10 and Fig. 11, the current optimal control parame-ters can be obtained within 20 cycles. Moreover, the rudder adjustment frequency based on MAD optimization is low, so it is more suitable for ships with large inertia.
2) TRACKING TIME-VARYING HEADING ANGLES (ON-LINE CONTROLLER)
For time-varying heading, the AMBPS algorithm is used to adjust the PID parameters on-line for tracking. Based on the optimum parameters n = 10, D0=0.99, Lstep =1, Vm = 5
obtained from SectionV-C1,ψreq=60sin(0.01t) is taken as
an example to simulate. Considering the efficiency problem, this paper takes [Kp, Ki, Kd] = [10, 0, 40] as the initial value to adjust the PID parameters. Besides, adjusting parameters
TABLE 7. Tracking error and the optimal PID parameters under different variable values (ψreq=60◦, minimum RMSE).
TABLE 8. Tracking different heading angle errors and PID parameters (AMBPS-PID, minimum RMSE).
TABLE 9. On-line adjustment steps and errors.
per step will also lead to inefficiency and may not be needed, so on-line adjustment can be divided into one-step adjustment and multi-steps adjustment. The PID parameters are adjusted at each TS step, and the error results are shown in Table9.
FIGURE 11. Tracking different heading angles (AMBPS-PID, minimum RMSE).
As can be seen from Table 9, whether minimum
MAD or RMSE as the standard, the tracking result is the best when TS = 130. Similarly, the error results for tracking other time-varying heading angles are shown in Table10, Fig.12
and Fig.13. Fig. 12a and Fig. 13a show the ship trajectories for different heading angles. Fig. 12b and Fig. 13b show the changes of PID parameters, the tracking heading angles, the minimum errors and the rudder angle, respectively. 3) PATH FOLLOWING SIMULATION RESULT
Path following control method adopts LOS navigation. Ran-dom path points are set for path following control. To make the tracking effect closer to the actual situation, and consid-ering the site area, the simulation environment was selected
FIGURE 12. Tracking different time-varying heading angles (AMBPS-PID, minimum MAD).
TABLE 10. Tracking different time-varying heading angles errors (ψreq=Asin(0.01t)).
from Kralingen Plas in Rotterdam, the Netherlands, as shown in Fig. 14a. Then set up the real target trajectory on Google Maps, as shown in Fig. 14b. Among them, path points include both clockwise and anticlockwise trajectories, which can fully reflect the path following effect. Based on SectionV-C1
and Section V-C2, the optimal parameters of AMBPS-PID algorithm are shown in Table11.
FIGURE 13. Tracking different time-varying heading angles (AMBPS-PID, minimum RMSE).
TABLE 11.Path following AMBPS-PID algorithm basic parameters.
Tracking the effect of manually adjusting the fixed param-eter values [KP, KI, KD] = [5, 0, 20] is shown in Fig. 15.
According to the on-line controller simulation analysis described in SectionV-C2, it can be concluded that the head-ing angle trackhead-ing result is optimal when the TS = 130. Since the essence of LOS navigation is to track the heading angle,
FIGURE 14. Map of Kralingen Plas in Rotterdam, The Netherlands.
FIGURE 15. Path following effect with manually adjusting parameters.
the path following also selects the parameter adjustment once every 130 steps. The tracking effect is shown in Fig. 16. To verify the correctness of the TS = 130 analysis, the
FIGURE 16. Path following effect with AMBPS-PID adjusting parameters.
TABLE 12.Path following errors under different TS.
tracking error results of different TS are given. The evaluation principle of tracking error is based on Eq.23, and the results are shown in Table12. From the results, it can be seen that the previous analysis is correct.
D. ANALYSIS AND COMPARISON OF RESULTS 1) RESULTS ANALYSIS
Section V-B and V-C describe the acquisition process of optimal parameters for adaptive motion control of 7m KVLCC2 ship. Through the Section V-B manual tuning, the simplest single heading angle is tested for preliminary judgment of parameter setting. For special case parameters, such as KI in this paper, the parameter value can be
deter-mined quickly and the follow-up research can be simplified. Based on the roughly estimated PID parameters, the tracking MAD and RMSE of single heading angle and time-varying angles can be obtained. Although the emphasis of MAD and
FIGURE 17. Tracking RMSE and MAD for ship control single heading angle and time-varying heading angles.
FIGURE 18. Comparison of manual setting parameters and AMBPS-PID online controller simulation results.
RMSE is different, the essence of MAD and RMSE is to reflect the errors between the calculated value and the target value, so the changing trend of the results is the same.
Section V-C shows how to search the optimal
con-troller parameters step by step, from a single heading angle to time-varying heading angles. When applied to the 7m KVLCC2 ship, the influence of AMBPS algorithm param-eters on control results is uncertain. To obtain the controller parameters suitable for this ship, optimal control parameters
of the AMBPS algorithm need to be first analyzed and obtained.
From Tables 5 and 7, it can be seen that the tracking error decreases with the increase of n, and does not change after n ≥ 10. These results showed that the search range became wider with the increase of population individuals. In the case of satisfying the results, to ensure the search efficiency, the minimum value of n satisfying the conditions is selected. However, D0and Lstephave little effect on tracking
TABLE 13. List of symbols.
errors, which indicates that these two parameters have little effect on the results. Particle velocity determines the direction and distance of particle search, and the error decreases to the same level with the increase of V . Through this series of analyses, the optimal parameters of AMBPS are obtained, and the optimal parameters are the same whether it is MAD optimal or RMSE optimal.
After obtaining the optimal AMBPS parameters, the head-ing control is further developed from a shead-ingle headhead-ing to time-varying heading angles, and an on-line controller is designed. On-line control needs to consider computational speed, which depends on the number of iterations and search frequency. The high search frequency will affect the actual navigation
FIGURE 19. Algorithms comparison results figure.
situation, and the low search frequency will affect the control accuracy. As shown in Fig.10, 11and Table9, the optimal PID parameters can be obtained within 20 iterations and the search frequency being 130 times. The final path following control is implemented based on the optimal control parame-ters obtained above. To visualize the effects of the algorithm, compare the simulation results, as described in SectionV-D2. 2) RESULTS COMPARISON
By summarizing the Tables2,3, 6, 8and 10, we can get an intuitive comparison chart between manual and AMBPS-PID adaptive adjustment of AMBPS-PID parameters for tracking errors of single heading and time-varying headings, as shown in Fig.17. From the comparison results, result errors of the AMBPS-PID algorithm are less than those of manual tuning. MAD is 0.19◦ and 0.03◦ lower on average, and RMSE is 0.05◦and 0.03◦lower on average.
Combining Fig.15and Fig.16, it can get the simulation results in the comparison chart of path following, as shown in Fig.18. From the enlarged part of Fig.18, it can be seen that the tracking result of AMBPS-PID is better than that of traditional PID at larger turning points. MAD is 0.35 m lower and RMSE is 0.68 m lower, which is closer to the target path. In summary, whether heading control or path following, the AMBPS-PID algorithm can achieve a better control effect. Because of the good control effect of the PID algorithm
TABLE 14. Test function list and results.
itself, the advantage of the AMBPS-PID algorithm is mainly reflected in these two aspects:
1) It does not need to recalculate the optimal PID param-eters when tracking different heading angles, so it is adaptive.
2) The error is significantly reduced at the larger turning points.
VI. SUMMARY
In this research, the AMBPS-PID algorithm is used to study the adaptive motion control of ships, and the head-ing and path followhead-ing controller are designed. Based on the simulation analysis, the conclusions of this paper are as follows:
1) The rudder turning rate control is introduced into the motion controller, which makes the control results more in line with the actual ship situation.
2) This paper combines BAS with the PSO algorithm and introduces an adaptive mutation operator. Based on this, the AMBPS algorithm is developed, and the fast search of PID optimal parameters is successfully realized.
3) AMBPS algorithm can optimize the parameters of PID with fewer iterations, which shows that the efficiency of this algorithm is better.
4) By choosing the best frequency of variation through simulation analysis, the adaptability and efficiency of the algorithm can be further realized, which has a cer-tain significance for actual navigation.
5) Whether heading control or path following, simula-tion results show that the error of the AMBPS-PID algorithm is smaller than that of traditional PID (whether MAD or RMSE). Among them, the most obvious advantage is at the larger turning points of path following.
In the future, this algorithm can be applied to the actual navigation test results. AMBPS algorithm itself can be further improved. Other algorithms can also be combined with the AMBPS algorithm to study ship motion control.
APPENDIXES APPENDIX A SYMBOLS
List of symbols is shown as Table13.
APPENDIX B
TEST FUNCTION LIST AND COMPARING RESULTS OF ALGORITHMS
Test function list and comparing results of algorithms are shown as Table14and Fig.19.
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LE WANG was born in Shijiazhuang, Hebei, China, in 1990. She received the B.S. and M.S. degrees in mechanical engineering from the Wuhan University of Technology, Wuhan, China, in 2014 and 2017, respectively. She is currently pursuing the Ph.D. degree. Her research interests include mechanical and logis-tics engineering, intelligent ship control, and optimization algorithm.
QING WU was born in Anhua, Hunan, China, in 1962. She received the M.S. degree in mechan-ical engineering from the Wuhan University of Technology, Wuhan, China, in 1997. Since 2005, she has been a Professor and a Ph.D. Supervi-sor with the Department of Logistics Automation, School of Logistics Engineering, Wuhan Univer-sity of Technology. She holds and participates in more than 30 projects. Her research interest includes water traffic safety and information.
JIALUN LIU was born in Fushun, Liaoning, China, in 1987. He received the M.S. degree in traf-fic information engineering and control from the Wuhan University of Technology, Wuhan, China, in 2013, and the Ph.D. degree in ship design, pro-duction and operation from the Delft University of Technology, Delft, The Netherlands, in 2017. He is currently an Associate Professor with the Intelligent Transportation Systems Research Cen-ter, Wuhan University of Technology. His research interests include the motion control of smart ships, (inland) ship maneuver-ability, and functional testing of smart ships.
SHIJIE LI was born in Jingmen, Hubei, China, in 1988. She received the M.S. degree in con-trol theory and engineering from Harbin Engineer-ing University, Harbin, China, in 2013, and the Ph.D. degree in transport engineering and logistics from the Delft University of Technology, Delft, The Netherlands, in 2016. Her research interests include the collaborative optimization of logistics systems, ship motion control, and multiship sys-tem operation optimization.
RUDY R. NEGENBORN received the Ph.D. degree from the Delft University of Technol-ogy, Delft, The Netherlands, in 2007. He is cur-rently a Full Professor of multimachine operations and logistics (full-time, fixed-term), the Head of section transport engineering and logistics, and the Director of studies MSC transport, infrastruc-ture, and logistics. He is the author of four books and more than one hundred articles. His research interests include logistics engineering and ship motion control.
